Select a particular rules out of a list

You can also use FilterRules:

rule = {beta -> 4, alpha -> 2, x -> 4, z -> 2, w -> 0.8};

FilterRules[rule, beta]
(* {beta -> 4} *)

FilterRules[rule, {beta, alpha}]
(* {beta -> 4, alpha -> 2} *)

Update: additional alternatives if you have V10:

KeyTake[rule,{alpha, x}]
(* or *) KeyTake[{alpha,x}][rule]
(* <|alpha->2,x->4|> *)
Normal@KeyTake[rule,{alpha, x}]
(*  {alpha->2,x->4} *)

KeySelect[rule, MatchQ[#,alpha|x]&]
(* or *) KeySelect[MatchQ[#,alpha|x]&][rule] 
(* <|alpha->2,x->4|>  *)
Normal@KeySelect[rule, MatchQ[#,alpha|x]&]
(*  {alpha->2,x->4} *)

Select[rule, MemberQ[{beta, alpha}, #[[1]]] &]

V9 style

If rule is a simple list of rules:

Cases[rule, _[beta | alpha, _]]

{beta -> 4, alpha -> 2}

---

other ways to go:

{beta, alpha} /. (# -> Rule[##] & @@@ rule)